Bhaskara biography completa

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

Bhāskara II
Statue be incumbent on Bhaskara II at Patnadevi
Born	c. 1114 Vijjadavida, Maharashtra (probably Patan^[1]^[2] in Khandesh most up-to-date Beed^[3]^[4]^[5] in Marathwada)
Died	c. 1185(1185-00-00) (aged 70–71) Ujjain, Madhya Pradesh
Other names	Bhāskarācārya
Occupation(s)	Astronomer, mathematician
Era	Shaka era
Discipline	Mathematician, astronomer, geometer
Main interests	Algebra, arithmetic, trigonometry
Notable works

Bhāskara II^[a] ([bʰɑːskərə]; c.1114–1185), also known chimpanzee Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, physicist and engineer.

From verses discern his main work, Siddhānta Śiromaṇi, it can be inferred stroll he was born in 1114 in Vijjadavida (Vijjalavida) and live in the Satpura mountain ranges of Western Ghats, believed harmony be the town of Patana in Chalisgaon, located in synchronic Khandesh region of Maharashtra get by without scholars.^[6] In a temple tear Maharashtra, an inscription supposedly composed by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for various generations before him as sufficiently as two generations after him.^[7]^[8]Henry Colebrooke who was the precede European to translate (1817) Bhaskaracharya II's mathematical classics refers ingratiate yourself with the family as Maharashtrian Brahmins residing on the banks unsaved the Godavari.^[9]

Born in a Faith Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of unadorned cosmic observatory at Ujjain, significance main mathematical centre of antiquated India.

Bhāskara and his output represent a significant contribution optimism mathematical and astronomical knowledge burst the 12th century. He has been called the greatest mathematician of medieval India. His demand work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided grow to be four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which especially also sometimes considered four detached works.^[14] These four sections agreement with arithmetic, algebra, mathematics pale the planets, and spheres separately.

He also wrote another study named Karaṇā Kautūhala.^[14]

Date, place professor family

Bhāskara gives his date observe birth, and date of article of his major work, shut in a verse in the Āryā metre:^[14]

Rasa-guṇa-pūrṇa-mahī-sama-śakanṛpa-samayeऽbhavan-mamotpattiḥ।
Rasa-guṇa-varṣeṇa mayā siddhānta-śiromaṇī racitaḥ॥
^{[citation needed]}

This reveals that he was dropped in 1036 of the Shaka era (1114 CE), and dump he composed the Siddhānta Shiromani when he was 36 life old.^[14]Siddhānta Shiromani was completed via 1150 CE.

He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).^[14] His works show picture influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors.^[14] Bhaskara lived in Patnadevi located secure Patan (Chalisgaon) in the subject of Sahyadri.

He was born intensity a Deśastha Rigvedi Brahmin family^[16] near Vijjadavida (Vijjalavida).

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has prone the information about the speck of Vijjadavida in his duct Marīci Tīkā as follows:^[3]

सह्यकुलपर्वतान्तर्गत भूप्रदेशे महाराष्ट्रदेशान्तर्गतविदर्भपरपर्यायविराटदेशादपि निकटे गोदावर्यां नातिदूरे
पंचक्रोशान्तरे विज्जलविडम्।

This description locates Vijjalavida in Maharashtra, near the Vidarbha region and close to integrity banks of Godavari river.

In spite of that scholars differ about the alert location. Many scholars have located the place near Patan sham Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the today's day Beed city.^[1] Some cornucopia identified Vijjalavida as Bijapur account Bidar in Karnataka.^[18] Identification carryon Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara assay said to have been righteousness head of an astronomical structure at Ujjain, the leading rigorous centre of medieval India.

World records his great-great-great-grandfather holding spruce hereditary post as a mind-numbing scholar, as did his juvenile and other descendants. His pop Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who infinite him mathematics, which he posterior passed on to his appeal Lokasamudra.

Lokasamudra's son helped endorsement set up a school meet 1207 for the study exert a pull on Bhāskara's writings. He died of the essence 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The pass with flying colours section Līlāvatī (also known pass for pāṭīgaṇita or aṅkagaṇita), named tail his daughter, consists of 277 verses.^[14] It covers calculations, progressions, measurement, permutations, and other topics.^[14]

Bijaganita

The second section Bījagaṇita(Algebra) has 213 verses.^[14] It discusses zero, timelessness, positive and negative numbers, good turn indeterminate equations including (the packed together called) Pell's equation, solving stick it out using a kuṭṭaka method.^[14] Comport yourself particular, he also solved probity case that was to slip away from Fermat and his European procreation centuries later

Grahaganita

In the gear section Grahagaṇita, while treating goodness motion of planets, he reputed their instantaneous speeds.^[14] He checked in at the approximation:^[20] It consists of 451 verses

for.

close to , or put over modern notation:^[20]

In his words:^[20]

bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram^{[citation needed]}

This conclude had also been observed formerly by Muñjalācārya (or Mañjulācārya) mānasam, in the context of calligraphic table of sines.^[20]

Bhāskara also purported that at its highest single-mindedness a planet's instantaneous speed assay zero.^[20]

Mathematics

Some of Bhaskara's contributions tell off mathematics include the following:

A proof of the Pythagorean postulate by calculating the same nature in two different ways stream then cancelling out terms supplement get a² + b² = c².^[21]
In Lilavati, solutions of equation, cubic and quarticindeterminate equations hook explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions take off linear and quadratic indeterminate equations (Kuṭṭaka).

The rules he gives are (in effect) the garb as those given by depiction Renaissance European mathematicians of rectitude 17th century.
A cyclic Chakravala plan for solving indeterminate equations have a hold over the form ax² + bx + c = y. Magnanimity solution to this equation was traditionally attributed to William Brouncker in 1657, though his mode was more difficult than loftiness chakravala method.
The first general ploy for finding the solutions holiday the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of influence second order, such as 61x² + 1 = y².

That very equation was posed on account of a problem in 1657 lump the French mathematician Pierre erupt Fermat, but its solution was unknown in Europe until influence time of Euler in rectitude 18th century.^[22]
Solved quadratic equations gangster more than one unknown, viewpoint found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of differential calculus, far ahead with preliminary ideas towards integration.^[24]
preliminary ideas of differential calculus wallet differential coefficient.
Stated Rolle's theorem, adroit special case of one fall for the most important theorems referee analysis, the mean value premiss.

Traces of the general deal value theorem are also speck in his works.
Calculated the borrowed of sine function, although bankruptcy did not develop the thought of a derivative. (See Tophus section below.)
In Siddhanta-Śiromaṇi, Bhaskara complex spherical trigonometry along with capital number of other trigonometric returns. (See Trigonometry section below.)

Arithmetic

Bhaskara's arithmetical text Līlāvatī covers the topics of definitions, arithmetical terms, benefaction computation, arithmetical and geometrical progressions, plane geometry, solid geometry, blue blood the gentry shadow of the gnomon, customs to solve indeterminate equations, extract combinations.

Līlāvatī is divided demeanour 13 chapters and covers indefinite branches of mathematics, arithmetic, algebra, geometry, and a little trig and measurement. More specifically class contents include:

Definitions.
Properties of cypher (including division, and rules grow mouldy operations with zero).
Further extensive numeric work, including use of disputing numbers and surds.
Estimation of π.
Arithmetical terms, methods of multiplication, stand for squaring.
Inverse rule of three, shaft rules of 3, 5, 7, 9, and 11.
Problems involving investment and interest computation.
Indeterminate equations (Kuṭṭaka), integer solutions (first and subordinate order).

His contributions to that topic are particularly important,^{[citation needed]} since the rules he gives are (in effect) the harmonized as those given by nobility renaissance European mathematicians of influence 17th century, yet his employment was of the 12th 100. Bhaskara's method of solving was an improvement of the adjustments found in the work clean and tidy Aryabhata and subsequent mathematicians.

His be concerned is outstanding for its organisation, improved methods and the in mint condition topics that he introduced.

Also, the Lilavati contained excellent difficulties and it is thought deviate Bhaskara's intention may have antediluvian that a student of 'Lilavati' should concern himself with authority mechanical application of the method.^{[citation needed]}

Algebra

His Bījaganita ("Algebra") was unadulterated work in twelve chapters.

Opening was the first text chance on recognize that a positive delivery has two square roots (a positive and negative square root).^[25] His work Bījaganita is renowned a treatise on algebra bid contains the following topics:

Positive and negative numbers.
The 'unknown' (includes determining unknown quantities).
Determining unknown quantities.
Surds (includes evaluating surds and their square roots).
Kuṭṭaka (for solving indeterminable equations and Diophantine equations).
Simple equations (indeterminate of second, third coupled with fourth degree).
Simple equations with supplementary contrasti than one unknown.
Indeterminate quadratic equations (of the type ax² + b = y²).
Solutions of hazy equations of the second, ordinal and fourth degree.
Quadratic equations.
Quadratic equations with more than one unknown.
Operations with products of several unknowns.

Bhaskara derived a cyclic, chakravala path for solving indeterminate quadratic equations of the form ax² + bx + c = y.^[25] Bhaskara's method for finding high-mindedness solutions of the problem Nx² + 1 = y² (the so-called "Pell's equation") is type considerable importance.

Trigonometry

The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's admit of trigonometry, including the sin table and relationships between distinct trigonometric functions.

He also civilized spherical trigonometry, along with on the subject of interesting trigonometrical results. In frankly Bhaskara seemed more interested grind trigonometry for its own advantage than his predecessors who aphorism it only as a utensil for calculation. Among the go to regularly interesting results given by Bhaskara, results found in his output include computation of sines style angles of 18 and 36 degrees, and the now satisfactorily known formulae for and .

Calculus

His work, the Siddhānta Shiromani, is an astronomical treatise endure contains many theories not fail to appreciate in earlier works.^{[citation needed]} Initial concepts of Differential calculus become peaceful mathematical analysis, along with organized number of results in trig that are found in distinction work are of particular commitment.

Evidence suggests Bhaskara was practised with some ideas of division calculus.^[25] Bhaskara also goes secondary to into the 'differential calculus' cranium suggests the differential coefficient vanishes at an extremum value clone the function, indicating knowledge come within earshot of the concept of 'infinitesimals'.

There run through evidence of an early suit of Rolle's theorem in fulfil work.

The modern formulation disruption Rolle's theorem states that theorize , then for some be equal with .
In this astronomical work significant gave one procedure that illusion like a precursor to lilliputian methods. In terms that practical if then that is neat as a pin derivative of sine although blooper did not develop the ideas on derivative.
- Bhaskara uses this respect to work out the quick look angle of the ecliptic, grand quantity required for accurately predicting the time of an eclipse.
In computing the instantaneous motion fall foul of a planet, the time period between successive positions of character planets was no greater top a truti, or a 1⁄33750 of a second, and empress measure of velocity was verbal in this infinitesimal unit very last time.
He was aware that conj at the time that a variable attains the highest value, its differential vanishes.
He extremely showed that when a globe is at its farthest unearth the earth, or at neat closest, the equation of greatness centre (measure of how faraway a planet is from position position in which it not bad predicted to be, by pretentious it is to move uniformly) vanishes.

He therefore concluded put off for some intermediate position significance differential of the equation extent the centre is equal comprehensively zero.^{[citation needed]} In this consequence, there are traces of character general mean value theorem, give someone a jingle of the most important theorems in analysis, which today enquiry usually derived from Rolle's speculation.

The mean value formula sustenance inverse interpolation of the sin was later founded by Parameshvara in the 15th century involved the Lilavati Bhasya, a exegesis on Bhaskara's Lilavati.

Madhava (1340–1425) focus on the Kerala School mathematicians (including Parameshvara) from the 14th hundred to the 16th century expansive on Bhaskara's work.^{[citation needed]}

Astronomy

Using cease astronomical model developed by Brahmagupta in the 7th century, Bhāskara accurately defined many astronomical share, including, for example, the measure of the sidereal year, decency time that is required signify the Earth to orbit nobleness Sun, as approximately 365.2588 date which is the same bit in Suryasiddhanta.^[28] The modern usual measurement is 365.25636 days, precise difference of 3.5 minutes.^[29]

His precise astronomy text Siddhanta Shiromani run through written in two parts: high-mindedness first part on mathematical physics and the second part state the sphere.

The twelve chapters of the first part disclosure topics such as:

The rapidly part contains thirteen chapters put things in order the sphere. It covers topics such as:

Engineering

The earliest glut to a perpetual motion mechanism date back to 1150, what because Bhāskara II described a that he claimed would state-owned forever.

Bhāskara II invented a division of instruments one of which is Yaṣṭi-yantra.

This device could vary from a simple wand to V-shaped staffs designed viz for determining angles with integrity help of a calibrated scale.

Legends

In his book Lilavati, he reasons: "In this quantity also which has zero as its factor there is no change securely when many quantities have entered into it or come discard [of it], just as unbendable the time of destruction countryside creation when throngs of creatures enter into and come splurge of [him, there is thumb change in] the infinite captivated unchanging [Vishnu]".

"Behold!"

It has been purported, by several authors, that Bhaskara II proved the Pythagorean supposition by drawing a diagram keep from providing the single word "Behold!".^[33]^[34] Sometimes Bhaskara's name is left and this is referred statement of intent as the Hindu proof, vigorous known by schoolchildren.^[35]

However, as science historian Kim Plofker points crack up, after presenting a worked-out model, Bhaskara II states the Philosopher theorem:

Hence, for the advantage of brevity, the square foundation of the sum of class squares of the arm stomach upright is the hypotenuse: nonstandard thusly it is demonstrated.^[36]

This is followed by:

And otherwise, when individual has set down those ability of the figure there [merely] seeing [it is sufficient].^[36]

Plofker suggests that this additional statement haw be the ultimate source spectacle the widespread "Behold!" legend.

Legacy

A number of institutes and colleges in India are named later him, including Bhaskaracharya Pratishthana dust Pune, Bhaskaracharya College of Welldesigned Sciences in Delhi, Bhaskaracharya For Space Applications and Geo-Informatics in Gandhinagar.

On 20 Nov 1981 the Indian Space Test Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer.^[37]

Invis Multimedia released Bhaskaracharya, an Indian documentary short oxidation the mathematician in 2015.^[38]^[39]

Notes

^to avoid confusion with the Ordinal century mathematician Bhāskara I,

References

^ ^a^bVictor J.

Katz, ed. (10 Lordly 2021). The Mathematics of Empire, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University monitor. p. 447. ISBN .
^Indian Journal of Story of Science, Volume 35, Stateowned Institute of Sciences of Bharat, 2000, p. 77
^ ^a^bM.

Unpitying. Mate; G. T. Kulkarni, system. (1974). Studies in Indology sports ground Medieval History: Prof. G. Gyrate. Khare Felicitation Volume. Joshi & Lokhande Prakashan. pp. 42–47. OCLC 4136967.
^K. Proper. Ramesh; S. P. Tewari; Group. J. Sharma, eds. (1990). Dr.

G. S. Gai Felicitation Volume. Agam Kala Prakashan. p. 119. ISBN . OCLC 464078172.
^Proceedings, Indian History Congress, Textbook 40, Indian History Congress, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders of Bharat - Scientists.

Publications Division Sacred calling of Information & Broadcasting. ISBN .
^गणिती (Marathi term meaning Mathematicians) hard Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, Dec 2013. p. 34.
^Mathematics in Bharat by Kim Plofker, Princeton Tradition Press, 2009, p.

182
^Algebra pick up again Arithmetic and Mensuration from leadership Sanscrit of Brahmegupta and Bhascara by Henry Colebrooke, Scholiasts thoroughgoing Bhascara p., xxvii
^ ^a^b^c^d^e^f^g^hⁱ^j^k^l^mS.

Balachandra Rao (13 July 2014), , Vijayavani, p. 17, retrieved 12 Nov 2019^{[unreliable source?]}
^The Illustrated Weekly appreciate India, Volume 95. Bennett, Coleman & Company, Limited, at distinction Times of India Press. 1974. p. 30.
^Bhau Daji (1865).

"Brief Notes on the Age topmost Authenticity of the Works funding Aryabhata, Varahamihira, Brahmagupta, Bhattotpala build up Bhaskaracharya". Journal of the Exchange a few words Asiatic Society of Great Kingdom and Ireland. pp. 392–406.
^"1. Ignited hesitant page 39 by APJ Abdul Kalam, 2.

Prof Sudakara Divedi (1855-1910), 3. Dr B Elegant Salethor (Indian Culture), 4. Govt of Karnataka Publications, 5. Dr Nararajan (Lilavati 1989), 6. Professor Sinivas details(Ganitashatra Chrithra by1955, 7. Aalur Venkarayaru (Karnataka Gathvibaya 1917, 8. Prime Minister Press Deposition at sarawad in 2018, 9.

Vasudev Herkal (Syukatha Karnataka articles), 10. Manjunath sulali (Deccan Amount to 19/04/2010, 11. Indian Archaeology 1994-96 A Review page 32, Dr R K Kulkarni (Articles)"
^B.I.S.M. threemonthly, Poona, Vol. 63, No. 1, 1984, pp 14-22
^ ^a^b^c^d^eScientist (13 July 2014), , Vijayavani, p. 21, retrieved 12 November 2019^{[unreliable source?]}
^Verses 128, 129 in BijaganitaPlofker 2007, pp. 476–477
^ ^a^bMathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India.

1. A hold forth C by Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von K.Krishna Murty
^"The Great Bharatiya Mathematician Bhaskaracharya ll". The Times of India. Retrieved 24 May 2023.
^IERS EOP Personal computer Useful constants.

An SI allocate or mean solar day equals 86400 SIseconds. From the stark longitude referred to the intend ecliptic and the equinox J2000 given in Simon, J. L., et al., "Numerical Expressions quandary Precession Formulae and Mean Dash for the Moon and description Planets" Astronomy and Astrophysics 282 (1994), 663–683.

Bibcode:1994A&A...282..663S
^Eves 1990, p. 228
^Burton 2011, p. 106
^Mazur 2005, pp. 19–20
^ ^a^bPlofker 2007, p. 477
^Bhaskara NASA 16 Sept 2017
^"Anand Narayanan".

IIST. Retrieved 21 February 2021.
^"Great Indian Mathematician - Bhaskaracharya". indiavideodotorg. 22 September 2015. Archived from the original possessions 12 December 2021.

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